Estimating Square and Cube Roots

Vocabulary and core mathematical procedures for estimating square and cube roots, determining whole-number proximity, and refining root approximations to one decimal place.

consecutive numbers

Whole numbers that follow each other, such as $3$ and $4$.

estimate

A careful guess that is close to the exact value.

midpoint

The number exactly halfway between two numbers.

approximately equal ($\approx$)

Nearly equal, but not exactly.

Estimating a square root between whole numbers

A method of identifying the perfect squares on either side of a number on a number line (for example, $9 < 11 < 16$, which gives $3 < \sqrt{11} < 4$).

Midpoint method for root proximity

Squaring the midpoint between two whole numbers to determine which number the root is closer to (for example, $3.5^2 = 12.25$, which is greater than $11$, showing $\sqrt{11} < 3.5$ and nearer to $3$).

Refining an estimate to one decimal place

Squaring tenths to bracket a root (e.g., $3.3^2 = 10.89$ and $3.4^2 = 11.56$) and testing the half-tenth midpoint ($3.35^2 = 11.2225 > 11$) to establish that $\sqrt{11} \approx 3.3$.